Untangling complex syste.., p.2
Untangling Complex Systems, page 2
7.3.2
Signal Transduction Systems with Positive Feedback ................... 177
7.4
Epigenetic Events ........................................................................................ 181
7.5
Biological Rhythms ..................................................................................... 185
7.6
Amplification and Adaptation in Regulatory and Sensory Systems ........... 187
7.6.1
Magnitude Amplification ............................................................... 187
7.6.2
Sensitivity Amplification ............................................................... 188
7.6.3 Adaptation ...................................................................................... 190
7.7
Key Questions ............................................................................................. 191
x
Contents
7.8
Key Words ................................................................................................... 192
7.9
Hints for Further Reading ........................................................................... 192
7.10 Exercises ...................................................................................................... 192
7.11
Solutions to the Exercises ............................................................................ 193
Chapter 8 The Emergence of Temporal Order in a Chemical Laboratory ............................... 197
8.1 Introduction ................................................................................................. 197
8.2
The Discovery of Oscillating Chemical Reactions ..................................... 197
8.3
The Systematic Design of Chemical Oscillators ......................................... 199
8.3.1 Excitability .....................................................................................202
8.3.2 Oscillations ....................................................................................203
8.3.3
In Practice ......................................................................................203
8.4
Primary “Oscillators” ..................................................................................205
8.4.1
Oregonator Model: The “Primary Oscillator” of Coproduct
Autocontrol ....................................................................................205
8.4.2
The Modified Lotka-Volterra or Predator-Prey “Primary
Oscillator” ......................................................................................208
8.4.3
The “Flow Control Primary Oscillator” ........................................ 212
8.4.4
The Composite System: A Chemical Equilibrium Coupled
to a “Primary Oscillator” ............................................................... 214
8.4.5
“Delayed Negative Feedback Oscillator” ...................................... 216
8.5
Overview and Hints for Further Reading ....................................................220
8.6
Key Questions ............................................................................................. 222
8.7
Key Words ................................................................................................... 222
8.8 Exercises ...................................................................................................... 222
8.9
Solutions to the Exercises ............................................................................ 227
Chapter 9 The Emergence of Order in Space ........................................................................... 241
9.1 Introduction ................................................................................................. 241
9.2
The Reaction-Diffusion Model ................................................................... 241
9.3
Turing Patterns ............................................................................................246
9.4
Turing Patterns in a Chemical Laboratory .................................................. 251
9.5
Turing Patterns in Nature ............................................................................ 255
9.5.1
Biology: The Development of Embryos ......................................... 256
9.5.2
Biology: Regeneration of Tissues...................................................260
9.5.3
Biology: Phyllotaxis ....................................................................... 261
9.5.4
Biology: Animal Markings ............................................................ 261
9.5.5
Ecology, Sociology, and Economy ................................................. 263
9.5.6 Geomorphology ............................................................................. 263
9.5.7
The Next Development of Turing’s Theory: The
Mechanochemical Patterning ........................................................264
9.6
Chemical Waves ..........................................................................................269
9.6.1
Propagator-Controller Model ......................................................... 270
9.6.1.1 Phase Waves ................................................................... 271
9.6.1.2 Trigger Waves ................................................................. 271
9.6.2
Shapes of Chemical Waves ............................................................ 273
9.6.2.1 Mono- and Bi-Dimensional Waves ................................ 273
9.6.2.2 Three-Dimensional Waves ............................................. 274
9.6.2.3 Effect of Curvature ......................................................... 274
Contents
xi
9.7
“Chemical” Waves in Biology ..................................................................... 275
9.7.1
Waves in a Neuron ....................................................................... 275
9.7.2
The Fisher-Kolmogorov Equation ................................................ 279
9.7.3
Waves in Our Brain ...................................................................... 281
9.7.4
Waves in Our Heart ...................................................................... 282
9.7.5
Calcium Waves ............................................................................. 283
9.7.6
cAMP Waves: The Case of Dictyostelium Discoideum...............284
9.7.7
Spreading of Species, Epidemics and … Fads .............................285
9.8
Liesegang Patterns.......................................................................................285
9.9
Liesegang Phenomena in Nature .................................................................288
9.9.1
In Geology ....................................................................................288
9.9.2
In Biology ..................................................................................... 289
9.10
A Final Note: The Reaction-Diffusion Structures in Art and
Technology .................................................................................................. 289
9.10.1 Reaction-Diffusion Processes as Art ........................................... 289
9.10.2 Reaction-Diffusion Processes in Technology ..............................290
9.11
Key Questions .............................................................................................290
9.12
Key Words ................................................................................................... 291
9.13
Hints for Further Reading ........................................................................... 291
9.14 Exercises ...................................................................................................... 291
9.15
Solutions to the Exercises ............................................................................296
Chapter 10 The Emergence of Chaos in Time ............................................................................ 317
10.1 Introduction ................................................................................................. 317
10.2
Nonlinearity and Chaos: The Case of the Double Pendulum ..................... 317
10.3
Nonlinearity and Chaos: The Case of the Population Growth and the
Logistic Map................................................................................................ 321
10.4
The Universality of Chaos ........................................................................... 326
10.5 Convection ................................................................................................... 327
10.6
The Entropy Production in the Nonlinear Regime: The Case of
Convection ................................................................................................... 332
10.7
The “Butterfly Effect” ................................................................................. 334
10.7.1 The Complexity of Convection in the Terrestrial Atmosphere ..... 334
10.7.2 The Lorenz’s Model ..................................................................... 335
10.7.3 The Sensitivity to the Initial Conditions ...................................... 338
10.7.4 The Hydrodynamic Photochemical Oscillator .............................340
10.8
Aperiodic Time Series ................................................................................. 342
10.8.1 How Do We Recognize Chaotic Time Series? ............................. 343
10.8.1.1 Time Delay τ ............................................................... 343
10.8.1.2 Embedding Dimension m ............................................344
10.8.1.3 Lyapunov Exponents ................................................... 345
10.8.1.4 Kolmogorov-Sinai Entropy .........................................346
10.8.1.5 Correlation Dimension ................................................ 347
10.8.1.6 Permutation Entropy ................................................... 347
10.8.1.7 Surrogate Data ............................................................348
10.8.1.8 Short-Term Predictability and Long-Term
Unpredictability ..........................................................348
10.8.2 Prediction of the Chaotic Time Series ......................................... 349
10.8.2.1 Artificial Neural Networks ......................................... 350
xii
Contents
10.9
Mastering Chaos........................................................................................ 352
10.9.1 Applications ............................................................................... 354
10.9.1.1 Communication by Chaotic Dynamics ..................... 354
10.9.1.2 Computing by Chaotic Dynamics............................. 354
10.10
Key Questions ........................................................................................... 355
10.11
Key Words ................................................................................................. 355
10.12
Hints for Further Reading ......................................................................... 356
10.13 Exercises .................................................................................................... 356
10.14
Solutions to the Exercises ..........................................................................360
Chapter 11 Chaos in Space: The Fractals ................................................................................... 379
11.1 Introduction ............................................................................................... 379
11.2
What Is a Fractal? ...................................................................................... 381
11.3
Fractal Dimension ..................................................................................... 383
11.4
Fractals That Are Not Perfectly Self-Similar............................................ 385
11.5
The Fractal-like Structures in Nature ....................................................... 386
11.6
The Dimensions of Fractals That Are Not Perfectly Self-Similar ............ 388
11.7
A Method for Generating Fractal-like Structures in the Lab .................... 389
11.8
Dendritic Fractals ...................................................................................... 392
11.9 Multifractals .............................................................................................. 394
11.9.1
Analysis of the Complex Images ............................................... 394
11.9.2
Analysis of the Complex Time Series ........................................ 395
11.10
Diffusion in Fractals ................................................................................. 395
11.11
Chemical Reactions on Fractals and Fractal-like Kinetics in Cells ......... 396
11.12
Power Laws or Stretched Exponential Functions? .................................... 399
11.13
Why Does Chaos Generate Fractals? ........................................................ 401
11.14
Chaos, Fractals, and Entropy ....................................................................402
11.15
Key Questions ...........................................................................................403
11.16
Key Words .................................................................................................403
11.17
Hints for Further Reading .........................................................................404
11.18 Exercises ....................................................................................................404
11.19
Solutions to the Exercises ..........................................................................405
Chapter 12 Complex Systems ..................................................................................................... 415
12.1
The Natural Complexity Challenges ......................................................... 415
12.2
The Computational Complexity of the Natural Complex Systems ........... 415
12.3
If It Were NP = P, Would Be the Complexity Challenges Surely Won? .. 419
12.4
The Features of Complex Systems ............................................................ 420
12.4.1 Networks .................................................................................... 420
12.4.2
Out-of-Equilibrium Systems ...................................................... 426
12.4.2.1 The Thermodynamics of Thermal Radiation ........... 426
12.4.2.2 The Fate of the Solar Thermal Radiation and the
Climate Change ........................................................ 430
12.4.2.3 Solar Radiation and Life on Earth ............................ 431
12.4.2.4 Solar Radiation as an Energy Source for Life on
Earth ......................................................................... 433
12.4.2.5 Solar Radiation as Information Source for Life
on Earth .................................................................... 437
12.4.3
Emergent Properties ...................................................................444
Contents
xiii
12.5
Key Questions ...........................................................................................446
12.6
Key Words .................................................................................................446
12.7
Hints for Further Reading ......................................................................... 447
12.8 Exercises .................................................................................................... 447
12.9
Solutions to the Exercises .......................................................................... 450
Chapter 13 How to Untangle Complex Systems? ....................................................................... 457
13.1 Introduction ............................................................................................... 457
13.2
Improving Electronic Computers .............................................................. 457
13.3
Natural Computing .................................................................................... 461
13.3.1 Computing Inspired by Natural Information Systems ................. 462
13.3.1.1
Artificial Life, Systems Chemistry, Systems
Biology, and Synthetic Biology ................................ 462
13.3.1.2
Membrane Computing ..............................................463
